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regularity

While delay differential equations with variable delays may have a superficial appearance of analyticity, it is far from clear in general that a global bounded solution $x(t)$ (namely, a bounded solution defined for all time $t$, such as a solution lying on an attractor) is an analytic function of $t$. Indeed, very often such solutions are not analytic, although they are often $C^\infty$. In this talk we provide sufficient conditions both for analyticity and for non-analyticity (but $C^\infty$ smoothness) of such solutions.

The current talk is concerned with transition fronts of nonlocal dispersal evolution equations in heterogeneous media. As it is known, solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal dispersal evolution equations. In the current talk, I will first present some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities.

I will describe results concerning the behavior of solutions of evolution equations with nonlocal dissipation and/or nonlocal forcing, including the Surface Quasi-Geostrophic equation and models of electroconvection.